# Phase noise

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In signal processing, phase noise is the frequency-domain representation of random fluctuations in the phase of a waveform, corresponding to time-domain deviations from perfect periodicity ("jitter"). Generally speaking, radio-frequency engineers speak of the phase noise of an oscillator, whereas digital-system engineers work with the jitter of a clock.

## Definitions

Historically there have been two conflicting yet widely used definitions for phase noise. Some authors define phase noise to be the spectral density of a signal's phase only, [1] while the other definition refers to the phase spectrum (which pairs up with the amplitude spectrum, see spectral density#Related concepts) resulting from the spectral estimation of the signal itself. [2] Both definitions yield the same result at offset frequencies well removed from the carrier. At close-in offsets however, the two definitions differ. [3]

The IEEE defines phase noise as ℒ(f) = Sφ(f)/2 where the "phase instability" Sφ(f) is the one-sided spectral density of a signal's phase deviation. [4] Although Sφ(f) is a one-sided function, it represents "the double-sideband spectral density of phase fluctuation". [5] [ clarification needed ] The symbol is called a (capital or uppercase) script L. [6]

## Background

An ideal oscillator would generate a pure sine wave. In the frequency domain, this would be represented as a single pair of Dirac delta functions (positive and negative conjugates) at the oscillator's frequency; i.e., all the signal's power is at a single frequency. All real oscillators have phase modulated noise components. The phase noise components spread the power of a signal to adjacent frequencies, resulting in noise sidebands. Oscillator phase noise often includes low frequency flicker noise and may include white noise.

Consider the following noise-free signal:

v(t) = Acos(2πf0t).

Phase noise is added to this signal by adding a stochastic process represented by φ to the signal as follows:

v(t) = Acos(2πf0t + φ(t)).

Phase noise is a type of cyclostationary noise and is closely related to jitter, a particularly important type of phase noise is that produced by oscillators.

Phase noise (ℒ(f)) is typically expressed in units of dBc/Hz, and it represents the noise power relative to the carrier contained in a 1 Hz bandwidth centered at a certain offsets from the carrier. For example, a certain signal may have a phase noise of −80 dBc/Hz at an offset of 10 kHz and −95 dBc/Hz at an offset of 100 kHz. Phase noise can be measured and expressed as single-sideband or double-sideband values, but as noted earlier, the IEEE has adopted the definition as one-half of the double-sideband PSD.

## Jitter conversions

Phase noise is sometimes also measured and expressed as a power obtained by integrating ℒ(f) over a certain range of offset frequencies. For example, the phase noise may be −40 dBc integrated over the range of 1 kHz to 100 kHz. This integrated phase noise (expressed in degrees) can be converted to jitter (expressed in seconds) using the following formula:

${\displaystyle {\text{jitter (seconds}})={\frac {{\text{phase error (}}{}^{\circ }{\text{)}}}{360^{\circ }\times {\text{frequency (hertz)}}}}}$

In the absence of 1/f noise in a region where the phase noise displays a –20 dBc/decade slope (Leeson's equation), the RMS cycle jitter can be related to the phase noise by: [7]

${\displaystyle \sigma _{c}^{2}={\frac {f^{2}{\mathcal {L}}\left(f\right)}{f_{\text{osc}}^{3}}}}$

Likewise:

${\displaystyle {\mathcal {L}}\left(f\right)={\frac {f_{\text{osc}}^{3}\sigma _{c}^{2}}{f^{2}}}}$

## Measurement

Phase noise can be measured using a spectrum analyzer if the phase noise of the device under test (DUT) is large with respect to the spectrum analyzer's local oscillator. Care should be taken that observed values are due to the measured signal and not the shape factor of the spectrum analyzer's filters. Spectrum analyzer based measurement can show the phase-noise power over many decades of frequency; e.g., 1 Hz to 10 MHz. The slope with offset frequency in various offset frequency regions can provide clues as to the source of the noise; e.g., low frequency flicker noise decreasing at 30 dB per decade (= 9 dB per octave). [8]

Phase noise measurement systems are alternatives to spectrum analyzers. These systems may use internal and external references and allow measurement of both residual (additive) and absolute noise. Additionally, these systems can make low-noise, close-to-the-carrier, measurements.

## Spectral purity

The sinewave output of an ideal oscillator is a single line in the frequency spectrum. Such perfect spectral purity is not achievable in a practical oscillator. Spreading of the spectrum line caused by phase noise must be minimised in the local oscillator for a superheterodyne receiver because it defeats the aim of restricting the receiver frequency range by filters in the IF (intermediate frequency) amplifier.

## Related Research Articles

Frequency modulation (FM) is the encoding of information in a carrier wave by varying the instantaneous frequency of the wave. The technology is used in telecommunications, radio broadcasting, signal processing, and computing.

In radio communications, single-sideband modulation (SSB) or single-sideband suppressed-carrier modulation (SSB-SC) is a type of modulation used to transmit information, such as an audio signal, by radio waves. A refinement of amplitude modulation, it uses transmitter power and bandwidth more efficiently. Amplitude modulation produces an output signal the bandwidth of which is twice the maximum frequency of the original baseband signal. Single-sideband modulation avoids this bandwidth increase, and the power wasted on a carrier, at the cost of increased device complexity and more difficult tuning at the receiver.

A crystal oscillator is an electronic oscillator circuit that uses the mechanical resonance of a vibrating crystal of piezoelectric material to create an electrical signal with a constant frequency. This frequency is often used to keep track of time, as in quartz wristwatches, to provide a stable clock signal for digital integrated circuits, and to stabilize frequencies for radio transmitters and receivers. The most common type of piezoelectric resonator used is the quartz crystal, so oscillator circuits incorporating them became known as crystal oscillators, but other piezoelectric materials including polycrystalline ceramics are used in similar circuits.

A phase-locked loop or phase lock loop (PLL) is a control system that generates an output signal whose phase is related to the phase of an input signal. There are several different types; the simplest is an electronic circuit consisting of a variable frequency oscillator and a phase detector in a feedback loop. The oscillator generates a periodic signal, and the phase detector compares the phase of that signal with the phase of the input periodic signal, adjusting the oscillator to keep the phases matched.

A signal generator is one of a class of electronic devices that generates electronic signals with set properties of amplitude, frequency, and wave shape. These generated signals are used as a stimulus for electronic measurements, typically used in designing, testing, troubleshooting, and repairing electronic or electroacoustic devices, though it often has artistic uses as well.

The power spectrum of a time series describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of a certain signal or sort of signal as analyzed in terms of its frequency content, is called its spectrum.

This is an index of articles relating to electronics and electricity or natural electricity and things that run on electricity and things that use or conduct electricity.

A spectrum analyzer measures the magnitude of an input signal versus frequency within the full frequency range of the instrument. The primary use is to measure the power of the spectrum of known and unknown signals. The input signal that most common spectrum analyzers measure is electrical; however, spectral compositions of other signals, such as acoustic pressure waves and optical light waves, can be considered through the use of an appropriate transducer. Spectrum analyzers for other types of signals also exist, such as optical spectrum analyzers which use direct optical techniques such as a monochromator to make measurements.

A variable frequency oscillator (VFO) in electronics is an oscillator whose frequency can be tuned over some range. It is a necessary component in any tunable radio transmitter or receiver that works by the superheterodyne principle, and controls the frequency to which the apparatus is tuned.

A voltage-controlled oscillator (VCO) is an electronic oscillator whose oscillation frequency is controlled by a voltage input. The applied input voltage determines the instantaneous oscillation frequency. Consequently, a VCO can be used for frequency modulation (FM) or phase modulation (PM) by applying a modulating signal to the control input. A VCO is also an integral part of a phase-locked loop. VCOs are used in synthesizers to generate a waveform whose pitch can be adjusted by a voltage determined by a musical keyboard or other input.

A Colpitts oscillator, invented in 1918 by American engineer Edwin H. Colpitts, is one of a number of designs for LC oscillators, electronic oscillators that use a combination of inductors (L) and capacitors (C) to produce an oscillation at a certain frequency. The distinguishing feature of the Colpitts oscillator is that the feedback for the active device is taken from a voltage divider made of two capacitors in series across the inductor.

Direct digital synthesis (DDS) is a method employed by frequency synthesizers used for creating arbitrary waveforms from a single, fixed-frequency reference clock. DDS is used in applications such as signal generation, local oscillators in communication systems, function generators, mixers, modulators, sound synthesizers and as part of a digital phase-locked loop.

In audio engineering, electronics, physics, and many other fields, the color of noise refers to the power spectrum of a noise signal. Different colors of noise have significantly different properties. For example, as audio signals they will sound different to human ears, and as images they will have a visibly different texture. Therefore, each application typically requires noise of a specific color. This sense of 'color' for noise signals is similar to the concept of timbre in music ; however, the latter is almost always used for sound, and may consider very detailed features of the spectrum.

A network analyzer is an instrument that measures the network parameters of electrical networks. Today, network analyzers commonly measure s–parameters because reflection and transmission of electrical networks are easy to measure at high frequencies, but there are other network parameter sets such as y-parameters, z-parameters, and h-parameters. Network analyzers are often used to characterize two-port networks such as amplifiers and filters, but they can be used on networks with an arbitrary number of ports.

Flicker noise is a type of electronic noise with a 1/f power spectral density. It is therefore often referred to as 1/f noise or pink noise, though these terms have wider definitions. It occurs in almost all electronic devices and can show up with a variety of other effects, such as impurities in a conductive channel, generation and recombination noise in a transistor due to base current, and so on.

A frequency synthesizer is an electronic circuit that generates a range of frequencies from a single reference frequency. Frequency synthesizers are used in many modern devices such as radio receivers, televisions, mobile telephones, radiotelephones, walkie-talkies, CB radios, cable television converter boxes, satellite receivers, and GPS systems. A frequency synthesizer may use the techniques of frequency multiplication, frequency division, direct digital synthesis, frequency mixing, and phase-locked loops to generate its frequencies. The stability and accuracy of the frequency synthesizer's output are related to the stability and accuracy of its reference frequency input. Consequently, synthesizers use stable and accurate reference frequencies, such as those provided by a crystal oscillator.

Compact Software was the first commercially successful microwave computer-aided design (CAD) company. The company was founded in 1973 by Les Besser to commercialize his eponymous program COMPACT, released when he was at Farinon Electric Company.

The concept of a linewidth is borrowed from laser spectroscopy. The linewidth of a laser is a measure of its phase noise. The spectrogram of a laser is produced by passing its light through a prism. The spectrogram of the output of a pure noise-free laser will consist of a single infinitely thin line. If the laser exhibits phase noise, the line will have non-zero width. The greater the phase noise, the wider the line. The same will be true with oscillators. The spectrum of the output of a noise-free oscillator has energy at each of the harmonics of the output signal, but the bandwidth of each harmonic will be zero. If the oscillator exhibits phase noise, the harmonics will not have zero bandwidth. The more phase noise the oscillator exhibits, the wider the bandwidth of each harmonic.

Oscillators produce various levels of phase noise, or variations from perfect periodicity. Viewed as an additive noise, phase noise increases at frequencies close to the oscillation frequency or its harmonics. With the additive noise being close to the oscillation frequency, it cannot be removed by filtering without also removing the oscillation signal.

Leeson's equation is an empirical expression that describes an oscillator's phase noise spectrum.

## References

1. Rutman, J.; Walls, F. L. (June 1991), "Characterization of frequency stability in precision frequency sources" (PDF), Proceedings of the IEEE, 79 (6): 952–960, Bibcode:1991IEEEP..79..952R, doi:10.1109/5.84972
2. Demir, A.; Mehrotra, A.; Roychowdhury, J. (May 2000), "Phase noise in oscillators: a unifying theory and numerical methods for characterization" (PDF), IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 47 (5): 655–674, CiteSeerX  , doi:10.1109/81.847872, ISSN   1057-7122
3. Navid, R.; Jungemann, C.; Lee, T. H.; Dutton, R. W. (2004), "Close-in phase noise in electrical oscillators", Proc. SPIE Symp. Fluctuations and Noise, Maspalomas, Spain
4. Vig, John R.; Ferre-Pikal, Eva. S.; Camparo, J. C.; Cutler, L. S.; Maleki, L.; Riley, W. J.; Stein, S. R.; Thomas, C.; Walls, F. L.; White, J. D. (26 March 1999), IEEE Standard Definitions of Physical Quantities for Fundamental Frequency and Time Metrology Random Instabilities, IEEE, ISBN   978-0-7381-1754-6, IEEE Std 1139-1999, see definition 2.7.
5. IEEE 1999 , p. 2, stating ℒ(f) "is one half of the double-sideband spectral density of phase fluctuations."
6. IEEE 1999 , p. 2
7. An Overview of Phase Noise and Jitter (PDF), Keysight Technologies, May 17, 2001
8. Cerda, Ramon M. (July 2006), "Impact of ultralow phase noise oscillators on system performance" (PDF), RF Design : 28–34